A crucial ingredient of our a posteriori analysis is a novel \(\mathbb {H}(\textbf {div})\) -conforming stress reconstruction obtained from the numerical solution by solving small problems on patches around mesh vertices.
In the spirit of [1]}, [2]}, we use the Arnold–Falk–Winther mixed finite element spaces with weakly enforced symmetry [3]}; strong symmetry could be enforced using the Arnold–Winther finite element spaces [4]} as in [1]}, but would come at a significantly higher computational cost.
The stress reconstruction is built so that its divergence and its normal component on the contact and Neumann portions of the domain boundary are locally in equilibrium with the volume and surface source terms, respectively (such equilibrium properties are not satisfied by stress fields resulting from \(H^1\) -conforming finite element approximations).
In order to distinguish the various error components, the stress reconstruction is additionally split so as to identify the contributions to the error resulting from linearization and regularization.
